I second not getting Apple Color composite monitors also but for very different reasons. I have 6 of them that dont work(AppleColor IIe, Apple color IIc, and the vanilla applecolor composite monitor), all almost identical in design.
FWIW, the last few Apple Color Composite monitors I've run into in the flesh (the last one was probably around 13 years ago now) still technically "worked" but they were universally in really sad shape. Discolored tubes, dim displays, etc. Given how these things were almost exclusively purchased by schools it's probably not at all surprising that they don't tend to be very well preserved.
(And no doubt because they were widely available as surplus they were kicking around for a long time as utility monitors for homebrew video production nerds; go to a sci-fi convention or something in the late 90's-early-2000's and you might see a guy pushing a cart with several of them stacked precariously on top of other random AV equipment. And I'm sure all that banging around didn't do them any good.)
Again, my recommendation if you want it to feel "oldschool" without investing much money in a potentially very flaky antique (and the use case is gaming, not "real work") would be a 90's 13" TV set with a composite input. Like anything CRT they are getting pretty thin on the ground, but you do still occasionally find a working one for cheap and it'll be *fine* for a 40 column machine like a ][plus.
(Unfortunately the main problem with this recommendation is you do pretty much have to luck out and find one at an old folk's garage sale or on a street corner for it to be cheap anymore, if someone's selling one on Craigslist or whatever the word has gotten out that these are desirable to GAMERs and they'll want a hundred bucks for it. I need to remember these things when I elect to *not* pick up the TV a neighbor leaves out on the curb because, well, that old junk should be worthless, something that seems to happen about once or twice a year.)
The VGA card suggestion isn't bad if you do want to use an LCD. The weird nonstandard NTSC video signal from Apple II's is... very hit or miss trying to display on the composite inputs of LCD TVs or via composite converters. A premium-brand TV with a good scaler
can make it look great, but your luck will very much vary. My living room TV looks nice with the II+ attached to it for games, but because the "color killer" circuitry Apple used to try to suppress the colorburst for text mode is bass-ackwards and doesn't really completely get rid of it it's a coin toss whether text will be an awful mess. And on the flip side it's not unusual for LCD TVs to not interpret Apple II video as being in color in all. Only way to know is to try it.